Ukraine's President Volodymyr Zelenskiy said negotiations were becoming "more realistic" while Russian Foreign Minister Sergei Lavrov said proposals now being discussed were "in my view close to an agreement".
The Kremlin said the sides were discussing status for Ukraine similar to that of Austria or Sweden, both members of the European Union that are outside the NATO military alliance.
Ukraine's chief negotiator said it was still demanding a ceasefire, the withdrawal of Russian troops and binding international security guarantees to protect Ukraine in future.
Though the war still ground on with Ukrainian civilians trapped in cities under Russian bombardment, the signs of compromise sent relief through global financial markets.
Shares in Germany - Russia's biggest energy market - were up 3.4 per cent.
In a speech to the US Congress that drew a long standing ovation, an unshaven Zelenskiy in an army green T-shirt appealed for tougher sanctions on Russia and more weapons to help his country fight "for the values of Europe and the world".
Speaking via a video link mainly in Ukrainian but closing in English, he invoked the Pearl Harbour and quoted Martin Luther King's "I have a dream" speech to call for a no-fly zone over Ukraine, although he acknowledged that the US has ruled this out.
"In the darkest time for our country, for the whole of Europe, I call on you to do more," he said.
Lawmakers were rapt as he played a video of clips showing dead and wounded children and buildings blasted by bombs.
Three weeks into the invasion, Russian troops have been halted at the gates of Kyiv, having taken heavy losses and failed to seize any of Ukraine's biggest cities in a war US officials say Russia expected to win within days.
Talks were due on Wednesday by a video link for a third straight day, the first time they have lasted more than a single day.
"The meetings continue and, I am informed, the positions during the negotiations already sound more realistic," Zelenskiy said in a video address overnight.
Later on Wednesday, he said Ukrainians must fight to "defend our state, our life, our Ukrainian life," but he also emphasised negotiations for "a just but fair peace for Ukraine, real security guarantees that will work."
In what was seen as a major shift, Zelenskiy had said on Tuesday Ukraine could accept international security guarantees that stopped short of its longstanding aim to join NATO.
Keeping Ukraine out of the military alliance was one of Russia's main demands in the months before it launched what it calls a "special operation" to disarm and "denazify" its neighbour.
"Neutral status is now being seriously discussed along, of course, with security guarantees," Russia's Lavrov said on Wednesday.
"There are absolutely specific formulations which in my view are close to agreement."
Vladimir Medinsky, Russia's chief negotiator, told state TV: "Ukraine is offering an Austrian or Swedish version of a neutral demilitarised state but at the same time a state with its own army and navy."
Kremlin spokesman Dmitry Peskov said the idea "could really be seen as a compromise".
Austria and Sweden, the biggest of six EU members outside NATO, both have small militaries that co-operate with the alliance.
Ukraine's chief negotiator, Mykhailo Podolyak, said guarantees were being discussed to provide "a rigid agreement with a number of guarantor states undertaking clear legal obligations to actively prevent attacks" on Ukraine.
He also said Ukraine was seeking direct talks between Zelenskiy and Russian President Vladimir Putin.
Russia has said they could meet but only to finalise an agreement already hammered out.
"Our position at the negotiations is quite specific - legally verified security guarantees; ceasefire; withdrawal of Russian troops. This is possible only with a direct dialogue between the heads of Ukraine and the Russian Federation," Podolyak tweeted.
Ukraine said about 20,000 people had escaped the besieged port of Mariupol in cars but hundreds of thousands remain trapped under bombardment without heat, power or water.
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